364 XV NORMED ALGEBRAS AND SPECTRAL THEORY sequence (xn)n^Q of elements of A, as follows: XQ = x, <p(0) = 0; suppose that <p(ri) has been defined and that the ek have been defined for 1 g k g (p(ri), and <p(») let xn = x - £ xefc. If jcn = 0, the sequence (ek) is finite and has (p(ri) elements; fc=i we take xm = xn = 0 for m ^ n, and <p(w) = <p(«) for m 2> «. If %M ^ 0, take a finite (possibly empty) sequence ODi^r of irreducible self-adjoint idempo- tents belonging to I, orthogonal in pairs and orthogonal to each ek for 1 ^ &<;<?(«), such that \\xe\\\2 £ \\x\\2/2n+\ and that the number r is as large as possible among all finite sequences having the above properties. (We have seen above that this number is <>2n+*.) Then put <p(« + 1) = <p(ri) + r; r ek = e\ for fc = cp(ri) + /, 1 <£ / ^ r; and xn+1 = xw - £ *<•. If no xn vanishes, i=l the sequence ((p(n)) tends to +00; for if it were bounded, we should have cp(m) = cp(ri) for some n and all m^n, and by definition this would mean that for each irreducible self-adjoint idempotent e' in the orthogonal supplement tp(n) F of £ Aek, we should have xe' = xne' = 0; but xn E F, xn ¥= 0, and F is a fc=i closed left ideal (15.8.2), so that this would contradict what was established at the beginning of the proof. This being so, it is clear from the construction that £ \\xen\\2 ^ \\x\\2 by n virtue of (15.8.9); hence if the series £ xen is not a finite sum, it converges in n any case to an element y e I which is the orthogonal projection of x on the closure of the left ideal a which is the sum of the Aen (for x — y, being m the limit of x— £ xek, is orthogonal to all the en) (6.5.2). If x — y ^0, *=i there would exist in the left ideal a1 an irreducible self-adjoint idempotent e" e I such that xe" -(x- y)e" ^ 0. If n is the smallest integer such that \\xe"\\2 ^ ||jc||2/2"+1, the existence of e" would contradict the maximality of the family of et such that <p(n) < i ^ cp(n + 1). Q.E.D. (15.8.11) Suppose that the algebra A is separable. Then every closed left ideal b is the Hilbert sum of a (finite or infinite) sequence of minimal left ideals ln = Aen, where en is an irreducible self-adjoint idempotent. For each x e b we have x = ^ xen, and for all x, y in b we have (x\y) = £ (xen \ yen). The second and third assertions are consequences of the first and of the definition of a Hilbert sum (6.4), since xen is the orthogonal projection of x on Aen (15.8.9). To prove the first assertion, let (xn)n^i be a dense sequence in b (3.10.9). We define inductively, for each «, a (finite or infinite) sequence (en, i)iein of irreducible self-adjoint idempotents, as follows. For (elti)ielnn A2||x»||- \\(x*?\\ <p(x*x)n for all